"Virasoro algebra"의 두 판 사이의 차이
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** (ii) discrete series (A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras) | ** (ii) discrete series (A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras) | ||
** (iii) GKO construction of discrete series (this is added on 22nd of Oct, 2007) | ** (iii) GKO construction of discrete series (this is added on 22nd of Oct, 2007) | ||
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* When is <math>M(c,h)</math> unitary? | * When is <math>M(c,h)</math> unitary? | ||
* to understand the submodules of the Verma module, we refer to Feigin and Fuks. | * to understand the submodules of the Verma module, we refer to Feigin and Fuks. | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">unitarity and ghost</h5> | ||
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+ | * unitarity means the inner product in the space of states is positive definite (or semi-positive definite) | ||
+ | * A state with negative norm is called a ghost. | ||
+ | * If a ghost is found on any level the represetation cannot occur in a unitary theory | ||
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<h5>character of minimal models</h5> | <h5>character of minimal models</h5> | ||
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+ | * [[minimal models]] | ||
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2010년 9월 1일 (수) 15:31 판
introduction
- Virasoro algebra could be pre-knowledge for the study of CFT
- important results on Virasoro algebra are
- (i)Kac Determinant Formula
- (ii) discrete series (A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras)
- (iii) GKO construction of discrete series (this is added on 22nd of Oct, 2007)
central charge and conformal weight
- \(c\) is called the central charge
- \(h\) is sometimes called a conformal dimension or conformal weights
Virasoro algebra
- Lie algebra of vector fields on the unit circle
\(f(z)\frac{d}{dz}\)
\(L_n=-z^{n+1}\frac{d}{dz}\) - they satisfy the following relation
\([L_m,L_n]=(m-n)L_{m+n}\) - taking a central extension of lie algebras, we get the Virasoro algebra
\(L_n,n\in \mathbb{Z}\)
\([c,L_n]=0\)
\([L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}\)
Verma module
- start with given c and h
- construct \(M(c,h)\)
- quotients from the Universal enveloping algebra
- tensor product from the one dimensional Borel subalgebra representations
- there exists a unique contravariant hermitian form
- contravariance means
- \(L_n\) and \(L_{-n}\) act as adjoints to each other, i.e.
\(<{L_n}v,w>=<w,L_{-n}w>\)
- \(L_n\) and \(L_{-n}\) act as adjoints to each other, i.e.
- a natural grading given by the \(L_0\)-eigenvalues
- contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight
- When is \(M(c,h)\) unitary?
- to understand the submodules of the Verma module, we refer to Feigin and Fuks.
unitarity and ghost
- unitarity means the inner product in the space of states is positive definite (or semi-positive definite)
- A state with negative norm is called a ghost.
- If a ghost is found on any level the represetation cannot occur in a unitary theory
unitary irreducible representations
- They are classified by c>1 and c<1 case.
- \(c> 1, h > 0\) positive definite
- \(c\geq 1, h \geq 0\) positive semi-definite
- \(0<c<1, h> 0\) with Kac determinant condition
- called the discrete series representations or minimal models
affine Lie algebras
- a highest weight representation V of an affine Kac-Moody algebra gives unitary representation of the Virasoro algebra
- This is because V is a unitary highest weight representation of the AKMA.
- Read chapter 4 of Kac-Raina on Wedge space
character of minimal models
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No-Ghost theorem
- refer to the No Ghost theorem
관련된 항목들
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Virasoro_algebra
- http://en.wikipedia.org/wiki/Coset_construction
- Virasoro algebra by V. Kac
articles
- Quantum Group Structure of the q-Deformed Virasoro Algebra
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- Haihong Hu
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- Unitary representations of the Virasoro and super-Virasoro algebras
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- P. Goddard, A. Kent and D. Olive, Comm. Math. Phys. 103, no. 1 (1986), 105–119.
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- Conformal invariance, unitarity and critical exponents in two dimensions
- Friedan, D., Qiu, Z. and Shenker, S., Phys. Rev. Lett. 52 (1984) 1575-1578
- Verma modules over the Virasoro algebra
- B.L. Feigin and D.B. Fuchs, Lecture Notes in Math. 1060 (1984), pp. 230–245.
- Infinite conformal symmetry in two-dimensional quantum field theory
- Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov, Nucl. Phys. B241 (1984) 333–380